A Bayesian Framework for Persistent Homology

An Introduction

Thomas Reinke

Baylor University

July 28, 2025

Introduction
Background
Bayesian Framework
Worked Example
Classification
References

Original Paper

A Bayesian Framework for Persistent Homology(Maroulas, Nasrin, and Oballe 2020)

Introduction

Real World Problem

Motivation
- Detecting crystalline structures of materials HEAs(High Entropy Alloys)

Current method APT(Atom Probe Tomography)
- 3D reconstruction of atoms
- Drawbacks - experimental noise and abundance of missing data - 65% of atoms in a sample are not registered in typical experiment & spacial coordinates of atoms are corrupted by external noise.
- Existing algorithms rely on symmetry arguments and cannot establish crystal lattice

Real World Problem: Body Centered Cubic

Real World Problem: Face Centered Cubic

TDA Motivation

Explore topological structure in datasets
- Persistent homolgy, associate shapes to data & summarize features with persistence diagrams

Applicable to many areas
- handwriting analysis, brain arteries, image analysis, neuroscience, sensor networks, protein structures, biology, dynamical systems, action recognition, signal analysis, chemistry, genetics, object data, etc.

TDA Motivation

Persistence Diagrams (PDs) for inference & classification problems

Model PDs as random sets(not vectors) - Poisson point processes - single parameter intensity
- Derive closed form of posterior - relies on conjugate families of Gaussian mixtures

Background

Example Data

Persistence Diagrams

Fundamental geometric structures

Simplices: Basic shapes like points, line segments, triangles, tetrahedra.

Simplicial Complexes: Collections of simplices “glued” together in a structured way.

Simplices: Geometric Building Blocks

A \(k\)-simplex is the convex hull of \(k+1\) geometrically independent points (vertices) \(\{v_0, ..., v_k\}\).
- Notation: \([v_0, ..., v_k] = \bigl\{ \sum_{i=0}^k \alpha_i v_i : \sum_{i=0}^k \alpha_i = 1, \alpha_i \ge 0 \bigr\}\)

Examples:
- 0-simplex: \([v_0]\) (a point/vertex)
- 1-simplex: \([v_0, v_1]\) (a line segment/edge)
- 2-simplex: \([v_0, v_1, v_2]\) (a triangle, including interior)
- 3-simplex: \([v_0, v_1, v_2, v_3]\) (a tetrahedron, including interior)

Faces: k-1 simplices spanned by subsets of the vertices. (e.g., edges of a triangle).

Simplicial Complexes

A Simplicial Complex \(S\) is a collection of simplices satisfying:

Face Inclusion: If a simplex \(\xi \in S\), then all faces of \(\xi\) are also in \(S\).

Consistent Intersection: The intersection of any two simplices \(\xi_1, \xi_2 \in S\) is either:
- Empty (\(\emptyset\)), or
- Contained in \(S\)

Intuition: A well-constructed mesh made of points, edges, triangles, etc.

Example

Vietoris-Rips

Goal: Build a simplicial complex from a point cloud \(X = \{x_i\}_{i=1}^L \subset \mathbb{R}^d\).
Vietoris-Rips Complex \(\mathcal{V}_r(X)\):
- Parameter: Proximity radius \(r > 0\).
- Rule: A simplex \([x_{i_1}, ..., x_{i_l}] \in \mathcal{V}_r(X)\) iff \(\text{diam} (x_{i_1}, ..., x_{i_l}) \leq r\)

Intuition: Connect groups of points that are close to each other.

VR Filtration

Given a nondecreasing sequence \(\{ r_n\} \in \mathbb{R}^+ \cup \{ 0\}\) with \(r_0 = 0\), we denote its Vietoris–Rips filtration by \(\{ \mathcal{V}_{r_n} (X)\}_{ n\in \mathbb{N}}\) .

A Vietoris-Rips Filtration is a sequence of nested simplicial complexes: \[ \mathcal{V}_{r_0}(X) \subseteq \mathcal{V}_{r_1}(X) \subseteq \mathcal{V}_{r_2}(X) \subseteq \dots \]

As we increase the radius \(r\), more points get connected, and higher-dimensional simplices appear.

This filtration tracks how the topological structure (connectivity, holes, voids) evolves as the scale \(r\) changes.

VR Visualization

Persistence Diagrams (PDs)

A Persistence Diagram (PD) \(\mathcal{D}\) is a multiset of points in the space \(\mathcal{W} = \mathbb{W} \times \{0, 1, ..., D-1\}\).
- The birth-death plane is \(\mathbb{W} = \{ (b, d) \in \mathbb{R}^2 \mid d \ge b \ge 0 \}\).

Each point \((b, d, k) \in \mathcal{D}\) represents:
- A homological feature of dimension \(k\).
- The feature appears (is born) at filtration scale \(b\).
- The feature disappears ( dies) at filtration scale \(d\).

Persistence: \(d-b\) (how long the feature “persists”). Longer persistence often indicates significant features.
Dimensions:
- \(k=0\): Connected components.
- \(k=1\): Loops or holes.
- \(k=2\): Voids or cavities.

Persistence Diagrams

The Poisson Point Process (PPP)

General Idea: Finite Point Processes \((\{p_n\}, \{\mathbb{P}_n\})\) model the number of points (\(N \sim \{p_n\}\)) and their spatial distribution (\((x_1, \dots, x_n) \sim \mathbb{P}_n\)).

Poisson PP \(\Pi\): A specific, widely used PP governed by an Intensity Measure \(\Lambda\) on \(\mathbb{X}\). Let \(\mu = \Lambda(\mathbb{X})\).
- Cardinality (\(N\)): The number of points \(N \sim \text{Poisson}(\mu)\).
- Spatial Distribution (\(\mathbb{P}_n\)): Given \(N=n\) points, their locations are drawn independently from the normalized intensity measure \(\frac{\Lambda(\cdot)}{\mu}\). \[ \mathbb{P}_n(A_1 \times \dots \times A_n) = \prod_{i=1}^n \left( \frac{\Lambda(A_i)}{\mu} \right) \]

Key Property: The intensity \(\Lambda(A)\) is the expected number of points falling within a region \(A \in \mathcal{X}\): \[ \mathbb{E}[|\Pi \cap A|] = \Lambda(A) \]

\(\mathbb{X}\) is polish space
\(\mathbb{P_n}\) symmetricy probablility measure on \(X^n\) which is the Borel \(\sigma\) algebra of \(\mathbb{X}\)
Why PPPs? We’re using PPPs because they provide a flexible framework for modeling point patterns like Persistence Diagrams, where both the number of points (topological features) and their locations (birth/death times) are random.
Intensity Measure \(\Lambda\): This is the core object defining a PPP.
- It’s a measure on the underlying space \(\mathbb{X}\).
- It controls where points are likely to appear. Regions \(A\) with higher \(\Lambda(A)\) are expected to contain more points.
- Often, \(\Lambda\) has a density \(\lambda(x)\) (the intensity function), so \(\Lambda(A) = \int_A \lambda(x) dx\). \(\lambda(x)\) represents the “rate” or “density” of points near \(x\).
- The total intensity \(\mu = \Lambda(\mathbb{X})\) is the expected total number of points in the process.
Cardinality: Unlike a fixed dataset, the total number of points \(N\) in a realization of a PPP is random, following a Poisson distribution with mean \(\mu\).
Spatial Distribution: This is crucial. For a given number of points \(n\), the PPP places them independently of each other. The probability of a point landing in a region \(A\) is proportional to \(\Lambda(A)\). Knowing a point exists at \(x\) gives no extra information about points nearby, beyond what \(\Lambda\) already tells us.
Key Property: This equation (\(\mathbb{E}[|\Pi \cap A|] = \Lambda(A)\)) directly connects the mathematical intensity measure \(\Lambda\) to a tangible meaning: the average number of points you’d find in any region \(A\). This justifies calling \(\Lambda\) the “intensity”.

Marked Poisson Point Processes

Motivation: Model points \(x \in \mathbb{X}\) that have associated attributes or ‘marks’ \(m\) from another space \(\mathbb{M}\). (e.g., PD points \(x\) with marks \(m\) describing feature properties).

Definition: A Marked PPP \(\Pi_M\) is a Point Process living on the product space \(\mathbb{X} \times \mathbb{M}\).

Key Components & Properties:
- Relies on an underlying standard PPP on \(\mathbb{X}\) (intensity \(\Lambda\)) to place the points \(x\).
- Uses a Stochastic Kernel \(\ell\): For any fixed location \(x \in \mathbb{X}\), \(\ell(x, \cdot)\) is a probability measure on the mark space \(\mathbb{M}\), defining the distribution \(P(\text{mark } | \text{ location } x)\).
- Given points \(\{x_1, \dots, x_n\}\), their marks \(\{m_1, \dots, m_n\}\) are generated conditionally independently, with \(m_i\) drawn from the distribution \(\ell(x_i, \cdot)\).

Intuition: Think of a two-stage process:
1. Generate point locations \(\{x_i\}\) in \(\mathbb{X}\) using a standard PPP.
2. For each generated point \(x_i\), independently generate its mark \(m_i\) using the probability distribution defined by the kernel \(\ell(x_i, \bullet)\).
\(\mathbb{M}\) is the dimension of the birth death points
Stochastic Kernel \(\ell\): This is the crucial link. For each specific location \(x\), \(\ell(x, \bullet)\) is a complete probability measure defined over all possible marks in \(\mathbb{M}\). It tells you the probability of getting any particular mark (or set of marks) given you are at location \(x\). This means \(\int_{\mathbb{M}} \ell(x, dm) = 1\).
Result: The final output \(\Pi_M\) is a collection of pairs \((x_i, m_i)\) living in the combined space \(\mathbb{X} \times \mathbb{M}\).
Usefulness: This structure allows modeling situations where point locations influence their properties, or where we might observe properties (marks) and want to infer locations (basis for later Bayesian section).

Bayesian Framework

Setting the Stage: Representation & Goal

Tilted Representation: We work with the “tilted” PD representation \((b, d-b)\) instead of \((b, d)\). This space is denoted \(\mathbb{W}\). (Notation Abuse: \(\mathcal{D}\) is used for the tilted version \(T(\mathcal{D})\)).

Focus: We analyze one homological dimension \(k\) at a time, assuming independence between dimensions (Model M1). (Notation Abuse: We write \(\mathcal{D}_X\) for \(\mathcal{D}^k_X\))

Goal: Develop a Bayesian framework to update our belief about a “true” underlying (latent) PD, represented by a prior intensity \(\lambda\), using one or more observed PDs \(\mathcal{D}_Y\).

Tilted Representation: First, a quick coordinate change. Instead of birth (\(b\)) and death (\(d\)), we’ll use birth (\(b\)) and persistence (\(p = d-b\)). This is just a linear transformation, often convenient for modeling. We denote the space of these \((b, p)\) points as \(\mathbb{W}\). We’ll often just write \(\mathcal{D}\) when we mean the set of these tilted points.
Focus on One Dimension (M1): A key simplifying assumption (Model M1 from the paper) is that topological features across different homological dimensions (e.g., \(H_0\) components vs. \(H_1\) loops vs. \(H_2\) voids) are statistically independent. This allows us to build our entire framework for one fixed dimension \(k\) at a time.
Notation Abuse: Because we’re focusing on a single, fixed \(k\) for the whole analysis (M1, M2, M3, Theorem 3.1), we’ll drop the subscript \(k\). So, when you see \(\mathcal{D}_X\) or \(\mathcal{D}_Y\), remember this implicitly means \(\mathcal{D}^k_X\) (the latent diagram for dimension \(k\)) and \(\mathcal{D}^k_Y\) (the observed diagram for dimension \(k\)).
The Goal: We aim to perform Bayesian inference. We start with a prior belief about the “true” underlying PD for our chosen dimension \(k\). This belief is encoded by modeling the latent PD, \(\mathcal{D}_X\), as a Poisson Point Process (PPP) with a spatial prior intensity \(\lambda(x)\). Then, we use observed data – specifically, one or more persistence diagrams \(\mathcal{D}_Y\) (if we have multiple datasets \(i=1, ..., m\), we’ll use \(D_{Y^1}, ..., D_{Y^m}\), all computed for the same dimension \(k\)) – to update our prior belief into a posterior intensity.

Bayesian Framework: Analogy

	Bayesian framework for RVs	Bayesian framework for random PDs
Prior	Modeled by a prior density \(f\)	Modeled by a Poisson PP with prior intensity \(\lambda\)
Likelihood	Depends on observed data	Stochastic kernel \(\color{purple}{\ell(y\|x)}\) depends on observed PDs
Posterior	Compute the posterior density	Defines a Poisson PP with posterior intensity

Core Idea: Treat unknown, latent PD \(\mathcal{D}^X\) as random draw from PPP defined by prior intensity \(\lambda_{\mathcal{D}^X}\). Use observed PDs \(D_{Y^1}, ..., D_{Y^m}\) (for fixed \(k\)) and likelihood kernel \(\color{purple}{\ell(y|x)}\) to get posterior intensity \(\lambda_{\mathcal{D}_X | D_{Y^{1:m}}}\).

Speaker Notes:

This table draws parallels between standard Bayesian inference (for parameters \(\theta\) of random variables) and our proposed framework for persistence diagrams (PDs). Remember, we’re focusing on one fixed homological dimension \(k\).
Prior Belief:
- Standard: We express prior belief about a parameter \(\theta\) using a probability density function \(f(\theta)\).
- PD Framework: Our prior belief about the latent PD \(\mathcal{D}^X\) (really \(\mathcal{D}^k_X\)) is encoded by modeling it as a Poisson Point Process (PPP). The key characteristic of this PPP is its prior intensity function \(\lambda(x)\) (short for \(\lambda_{\mathcal{D}^k_X}(x)\)). This function tells us the expected density of “true” latent features at location \(x=(b, p)\) in the persistence plane \(\mathbb{W}\).
Data Connection (Likelihood):
- Standard: The likelihood function \(L(\text{data}|\theta)\) quantifies how probable the observed data is, given a specific parameter value \(\theta\).
- PD Framework: The link between the latent features \(x\) (from \(\mathcal{D}^k_X\)) and the observed features \(y\) (in \(\mathcal{D}^k_Y\)) is defined by a stochastic kernel \(\color{purple}{\ell(y|x)}\). This kernel acts like our likelihood: it gives the probability density of observing a feature \(y\) assuming it originated from a specific latent feature \(x\). The actual data we use are the points in the observed diagram(s) \(D_{Y^1}, \dots, D_{Y^m}\). (This corresponds to the purple term in the theorem).
Updated Belief (Posterior):
- Standard: Combining the prior \(f(\theta)\) and likelihood \(L(\text{data}|\theta)\) via Bayes’ theorem gives the posterior density \(f(\theta|\text{data})\).
- PD Framework: Combining the prior intensity \(\lambda(x)\) and the likelihood kernel \(\color{purple}{\ell(y|x)}\) using the observed data \(D_{Y^{1:m}} = \cup_{i=1}^m D_{Y^i}\) yields the posterior intensity function \(\lambda(x | D_{Y^{1:m}})\). This function defines a posterior Poisson Point Process, which represents our updated probabilistic belief about the latent PD \(\mathcal{D}^k_X\) after seeing the data. (The formula involves the red, blue, purple, and darkgreen components).
Core Idea Summary: We model the unknown true PD (for dimension \(k\)) as a PPP realization governed by a prior intensity \(\lambda\). We use the actual points from observed diagrams (\(D_{Y^1}, \dots, D_{Y^m}\), all for dim \(k\)) and a kernel \(\color{purple}{\ell(y|x)}\) (our likelihood) to compute a posterior intensity. This posterior intensity defines the PPP representing our updated understanding of the true PD structure.

The Model: Latent PD \(\mathcal{D}_X\)

The “true” underlying PD \(\mathcal{D}_X\) is modeled as a Poisson Point Process (PPP) with a prior intensity density \(\lambda_{\mathcal{D}_X}(x)\).

Any potential feature \(x\) from the prior might be observable or vanish (unobserved).
- \(\alpha(x)\): Probability that feature \(x\) is potentially observable.
\(\mathcal{D}_X\) is decomposed into two independent PPPs:
- \(\mathcal{D}_{XO}\) (Observed Track): Points that could generate an observation.
  - Intensity: \(\color{blue}{\alpha(x) \lambda_{\mathcal{D}_X}(x)}\)
- \(\mathcal{D}_{XV}\) (Vanished Track): Points that are missed / not observed.
  - Intensity: \(\color{red}{(1-\alpha(x)) \lambda_{\mathcal{D}_X}(x)}\)

This slide describes Model M2, which defines our prior belief about the latent (“true”) persistence diagram, \(\mathcal{D}^X\).
Remember the Context (M1): We are operating under Model M1, meaning we’ve fixed a specific homological dimension \(k\). So, \(\mathcal{D}_X\) here is technically \(\mathcal{D}^k_X\), representing the true features only for that dimension \(k\).
Prior Model: We model this latent \(\mathcal{D}_X\) as a Poisson Point Process (PPP) on the persistence plane \(\mathbb{W}\). The behavior of this PPP is governed by its prior intensity density \(\lambda_{\mathcal{D}_X}(x)\) (really \(\lambda_{\mathcal{D}^k_X}(x)\)). This function reflects our prior assumptions about the density of true dimension-\(k\) features at location \(x=(b, p)\).
Detectability \(\alpha(x)\): We acknowledge that not all true features might be detectable in real data. We introduce \(\alpha(x)\), the probability that a true feature at location \(x\) is potentially observable. Conversely, \(1-\alpha(x)\) is the probability that it vanishes (is missed entirely, perhaps due to low persistence, noise, or sampling issues). \(\alpha(x)\) models feature detectability.
Decomposition: Using \(\alpha(x)\), we can think of the original latent process \(\mathcal{D}^X\) as being composed of two independent PPPs (this is a standard PPP property called “thinning”):
- \(\mathcal{D}_{XO}\) (Observable Track): The subset of latent features that could potentially generate an observation in our data. Its intensity is the prior intensity modulated by the probability of being observable: \(\color{blue}{\alpha(x) \lambda_{\mathcal{D}_X}(x)}\). This is the “signal” part of our prior that can interact with data (related to the blue term in the theorem’s denominator).
- \(\mathcal{D}^{XV}\) (Vanished Track): The subset of latent features that are fundamentally unobservable. Its intensity is modulated by the probability of vanishing: \(\color{red}{(1-\alpha(x)) \lambda_{\mathcal{D}_X}(x)}\). This part of our prior belief cannot be updated by observations (this is the red term in the theorem).
Consistency: Note that the intensities sum correctly: \(\color{blue}{\alpha(x)\lambda_{\mathcal{D}_X}(x)} + \color{red}{(1-\alpha(x))\lambda_{\mathcal{D}_X}(x)} = \lambda_{\mathcal{D}_X}(x)\), recovering the total prior intensity.

The Model: Observed PD \(\mathcal{D}_Y\)

An observed PD \(\mathcal{D}_Y\) is also decomposed into two independent components:

\(\mathcal{D}_{YO}\) (Observed from Signal): Points generated from the latent observable points \(\mathcal{D}_{XO}\).
- The pair \((\mathcal{D}_{XO}, \mathcal{D}_{YO})\) forms a Marked PPP.
- Connection is via the stochastic kernel \(\color{purple}{\ell(y|x)}\) (the “likelihood”). It gives the probability density of observing \(y \in \mathcal{D}_{YO}\) given a latent point \(x \in \mathcal{D}_{XO}\).
\(\mathcal{D}_{YS}\) (Spurious / Noise): Points arising independently from noise, clutter, or unanticipated geometry.
- Modeled as an independent PPP with intensity \(\color{darkgreen}{\lambda_{\mathcal{D}_{YS}}(y)}\).

Now we turn to Model M3, which describes the structure of the observed persistence diagrams \(\mathcal{D}_Y\).
Context: Again, this is for our fixed dimension \(k\). So \(\mathcal{D}_Y\) means \(\mathcal{D}^k_Y\). Crucially, \(\mathcal{D}_Y\) represents the random process that generates the actual diagrams \(D_{Y^1}, \dots, D_{Y^m}\) which we compute from our data. M3 models the underlying mechanism producing these diagrams.
Decomposition: The model assumes that any observed diagram (a realization of \(\mathcal{D}_Y\)) is composed of points from two independent sources:
- 1. \(\mathcal{D}_{YO}\) (Observed from Signal): These are the points in the observed diagram that correspond to the “true”, potentially observable latent features we defined in M2 (the points in \(\mathcal{D}_{XO}\)).
  - Marked PPP: The link between the latent signal \(\mathcal{D}_{XO}\) and the observed signal \(\mathcal{D}_{YO}\) is modeled as a Marked Poisson Point Process. Think of it this way: nature selects a latent feature \(x\) to be potentially observable (it’s in \(\mathcal{D}_{XO}\)). Then, the observation process generates a corresponding observed feature \(y\) (which becomes a point in \(\mathcal{D}_{YO}\)) according to some probability distribution. The observed point \(y\) is the “mark” associated with the latent point \(x\).
  - Stochastic Kernel \(\color{purple}{\ell(y|x)}\): This kernel defines the probability density of the mark \(y\) given the original latent point \(x\). This \(\color{purple}{\ell(y|x)}\) acts as our likelihood function (the purple term in the theorem). It models how the process of generating data and computing the PD might shift or perturb a true feature’s location from \(x\) to \(y\).
- 1. \(\mathcal{D}_{YS}\) (Spurious / Noise): These are points appearing in the observed diagram that do not originate from any true latent feature in \(\mathcal{D}_X\). They arise independently from sources like measurement noise, computational artifacts, sampling variations, or other unmodeled aspects.
  - This noise component is modeled as an independent PPP governed by its own intensity function \(\color{darkgreen}{\lambda_{\mathcal{D}_{YS}}(y)}\). This intensity describes the expected spatial density of purely spurious points appearing near \(y\) in an observed diagram. It’s our background noise model for dimension \(k\) (the darkgreen term in the theorem).
Interpretation: According to M3, any specific point \(y\) that we see in one of our actual computed diagrams (say, \(D_{Y^i}\)) could have either been generated by some latent feature \(x\) (via \(\color{purple}{\ell(y|x)}\)) or it could be purely noise (generated according to \(\color{darkgreen}{\lambda_{\mathcal{D}_{YS}}(y)}\)). The Bayesian theorem will weigh these possibilities.
(Refer to Figure 4 if available): Figure 4 visually summarizes these relationships: \(\mathcal{D}_X\) (latent) splits into \(\mathcal{D}_{XO}\) (latent signal) and \(\mathcal{D}_{XV}\) (latent vanished). \(\mathcal{D}_Y\) (observed process) splits into \(\mathcal{D}_{YO}\) (observed signal) and \(\mathcal{D}_{YS}\) (observed noise). Only \(\mathcal{D}_{XO}\) and \(\mathcal{D}_{YO}\) are directly linked via the likelihood kernel \(\color{purple}{\ell(y|x)}\).

Bayes’ Theorem for PDs

Goal: Find the posterior intensity \(\lambda_{\mathcal{D}_X | D_{Y^{1:m}}}(x)\) for the latent PD \(\mathcal{D}_X\), given \(m\) independent observed PDs \(D_{Y^1}, \dots, D_{Y^m}\).
Let \(D_{Y^{1:m}} = \cup_{i=1}^m D_{Y^i}\). The posterior intensity is:

\[ \lambda_{\mathcal{D}_X | D_{Y^{1:m}}}(x) = \underbrace{\color{red}{(1 - \alpha(x))\lambda_{\mathcal{D}_X}(x)}}_{\text{Prior Vanished Part}} + \underbrace{\frac{1}{m} \alpha(x) \sum_{i=1}^m \sum_{y \in D_{Y^i}} \frac{ \color{purple}{\ell(y|x)}\color{blue}{ \lambda_{\mathcal{D}_X}(x)}}{\color{darkgreen}{\lambda_{\mathcal{D}_{Y_S}}(y)} + \color{blue}{\int_{\mathbb{W}} \ell(y|u) \alpha(u) \lambda_{\mathcal{D}_X}(u) du}}}_{\text{Update from Observed Points } y} \quad \text{a.s.} \]

This formula gives the posterior intensity \(\lambda_{\mathcal{D}_X | D_{Y^{1:m}}}(x)\), which represents our updated belief (as an expected density) about the presence of a latent feature at location \(x \in \mathbb{W}\), given the observed data \(D_{Y^{1:m}}\). It combines the prior belief with information from the observations. Let’s break it down:

Overall Structure: Posterior Intensity = Prior Vanished Part + Update from Observations
Term 1: Prior Vanished Part \(\color{red}{(1 - \alpha(x)) \lambda_{D_X(x)}}\)
- \(\lambda_{\mathcal{D}_X}(x)\): Prior intensity density at \(x\) (from M2).
- \(\alpha(x)\): Probability that a feature at \(x\) is potentially observable (from M2).
- \((1 - \alpha(x))\): Probability that a feature at \(x\) is inherently unobservable (vanished, part of \(\mathcal{D}^{XV}\)).
- This entire term \((\color{red}{red})\) represents the contribution from the part of the prior belief (\(\mathcal{D}^{XV}\)) that can never be informed by observations. It passes directly to the posterior unchanged.
Term 2: Update from Observed Points \(y\) (\(\frac{1}{m} \sum ...\) part)
- This term adjusts the prior belief about potentially observable features based on the actual observations \(D_{Y^{1:m}}\).
- \(\frac{1}{m}\): Averages the update contribution across the \(m\) independent observed diagrams.
- \(\sum_{i=1}^m \sum_{y \in D_{Y^i}}\): Sums the influence of every observed point \(y\) from all \(m\) observed diagrams \(D_{Y^i}\).
- The Fraction: This core part weights the influence of each observed \(y\) on the posterior belief at \(x\).
  - Numerator \(\alpha(x) \ell(y|x) \lambda_{\mathcal{D}_X}(x)\): (Note: \(\alpha(x)\) is outside sum in formula, but conceptually linked here)
    - \(\color{blue}{\alpha(x) \lambda_{D_X(x)}}\): This is the prior expected density of observable features specifically at location \(x\). (From M2).
    - \(\color{purple}{\ell(y|x)}\): The likelihood kernel (from M3). How well does latent \(x\) explain observed \(y\)? (\(\color{purple}{purple}\))
    - The numerator combines the prior belief about observable features at \(x\) (via \(\alpha(x)\) and \(\lambda_{\mathcal{D}_X}(x)\)) with the likelihood of \(y\) given \(x\).
  - Denominator: Represents the total expected density of observing a point at location \(y\), considering all possibilities:
    - \(\color{darkgreen}{\lambda_{D_{Y_S}}(y)}\): Intensity of the spurious/noise process at \(y\) (from M3). How likely is \(y\) to be just noise? ()
    - \(\color{blue}{\text{integral over } W \text{ of } \ell(y|u) \alpha(u) \lambda_{D_X}(u) du}\): The marginal density of observing \(y\) generated from any potentially observable latent feature \(u \in \mathcal{D}^{XO}\). The term inside represents the prior expected density of observable features at a generic location u. (\(\color{blue}{ \int}\))
  - The fraction represents the contribution of observation \(y\) to updating the belief at \(x\), normalized by the total evidence for \(y\).
- Putting the Update Term Together: The averaged sum (\(\frac{1}{m} \sum \sum ...\), multiplied by \(\alpha(x)\)) combines the weighted evidence from all observations \(y\).
What is the Posterior? The result \(\lambda_{\mathcal{D}_X | D_{Y^{1:m}}}(x)\) is the intensity function of a new Poisson Point Process. This PPP represents our updated probabilistic model for the latent (“true”) persistence diagram \(\mathcal{D}_X\) after observing the data \(D_{Y^{1:m}}\).
Why “almost surely” (a.s.)?
- This theorem describes a relationship between random objects. The observed diagrams \(D_{Y^1}, \dots, D_{Y^m}\) are realizations of random point processes (\(\mathcal{D}_Y\)). Consequently, the posterior intensity \(\lambda_{\mathcal{D}_X | D_{Y^{1:m}}}(x)\) is itself a random function because its value depends on this random data.
- “Almost surely” is a standard term in probability theory meaning the equality holds with probability 1. It asserts that for any specific set of observations \(D_{Y^{1:m}}\) you might get (excluding zero-probability outcomes), this formula correctly describes the posterior intensity.

Interpretation & Computation: Gaussian Mixtures

Posterior: \(\lambda_{\mathcal{D}_X | D_{Y^{1:m}}}(x)\) blends prior belief (observable & unobservable) with updates from observed \(y\) via likelihood \(\color{purple}{\ell(y|x)}\) relative to noise.

Key Idea: Likelihood \(\color{purple}{\ell(y|x)}\) acts on PD points (\(x, y \in \mathbb{W}\)), avoiding raw data issues. But posterior calculation (Thm 3.1) can be complex.

Solution: Conjugate Priors via Gaussian Mixtures (GMs)
- Use Gaussian Mixtures (GMs) restricted to \(\mathbb{W}\) (\(\mathcal{N}^*\)) for conjugacy (posterior stays in the same family).
- Assume (M2’, M3’): Prior \(\lambda_{\mathcal{D}_X}\), Likelihood \(\color{purple}{\ell(y|x)}\) (single \(\mathcal{N}^*\)), Noise \(\color{darkgreen}{\lambda_{\mathcal{D}_{YS}}}\) are Gaussian/GM based.
- Result (Prop 3.2): Posterior \(\lambda_{\mathcal{D}_X | D_{Y^{1:m}}}(x)\) is also a GM.
- Benefit: Parameters update via standard Gaussian rules \(\implies\) computationally tractable framework.

Quick Recap: The posterior intensity \(\lambda_{\mathcal{D}_X | D_{Y^{1:m}}}(x)\) is our updated belief. It merges the prior with evidence from observed points \(y\), weighted by the likelihood \(\color{purple}{\ell(y|x)}\) and noise levels (\(\color{darkgreen}{\lambda_{\mathcal{D}_{YS}}}\) and the integrated signal part).
Core Concept: We work directly with likelihoods \(\color{purple}{\ell(y|x)}\) between PD points, which is conceptually powerful and avoids raw data problems. However, the posterior formula isn’t always simple to compute.
The Fix: Conjugacy with Gaussian Mixtures: To make this practical, we use a conjugate prior approach with Gaussian Mixtures (GMs), restricted to the valid PD wedge (\(\mathcal{N}^*\)). Conjugacy means the posterior will also be a GM if the prior is.
- Setup: We assume the prior intensity (\(\lambda_{\mathcal{D}_X}\)), the likelihood kernel (\(\color{purple}{\ell(y|x)}\)), and the noise intensity (\(\color{darkgreen}{\lambda_{\mathcal{D}_{YS}}}\)) are all built from these restricted Gaussians/GMs.
- Result: The math works out (Proposition 3.2) – the posterior intensity is guaranteed to be a GM as well.
Punchline: This is great because updating the GM parameters (weights, means, variances) follows standard, computable rules based on Gaussian properties. It makes the Bayesian inference framework efficient.

Worked Example

Parameters Summary

Data Generation Parameters
Case	Data Noise Std Dev (σ_Data)
Case I	0.032
Case II	0.100
Case III	0.316

Simulated Point Clouds

Prior Intensities

Case I Results

Case II Results

Case III Results

Case IV Results

Classification

Real-World Example: Classifying Crystal Lattices

Problem:

Classify Body-Centered Cubic (BCC) vs. Face-Centered Cubic (FCC) crystal structures in high-entropy alloys.
Data is from Atom Probe Tomography (APT): inherently noisy and sparse (~65% missing data).
Existing methods struggle with combined noise/sparsity or require prior knowledge of global structure.

Approach using Persistent Homology (PH):

Represent spatial atom configurations as point clouds.
Compute Rips filtrations and extract 1-dimensional Persistence Diagrams (PDs).
Idea: Topological features (connectedness, holes/voids) captured by PDs differ between BCC (center atom) and FCC (center void, face atoms).
- (See Figure 10 in paper for example PDs from each class).

Bayesian Classification: Method & Results

Experiment Setup:

Data: 200 BCC PDs, 200 FCC PDs.
Priors Tested :
- Prior-1: Class-specific (means from K-means on subset of data).
- Prior-2: Common “flat” prior (mean=(1,1), \(\sigma= \sqrt{20}\), high variance).
Likelihood/Noise Params:
- Likelihood kernel std dev: \(\sigma_{K} = 0.1\) (referred to as \(\sigma_{D_{YO}}\) in text).
- Unexpected noise intensity: \(\lambda_{D_{YS}}(x) = 5 \mathcal{N}^*(x; (0, 0), \sigma=\sqrt{0.2})\). (High weight=5 reflects noise features are rare).
Evaluation: 10-fold Cross-Validation, Area Under ROC Curve (AUC), Bootstrapped 95% CIs.

Bayesian Classification: Method & Results

Results (AUC):

10-Fold CV AUC Summary (Bootstrapped)
Prior	Mean AUC	5th Percentile	95th Percentile
Prior-1 (Class-Specific)	0.941	0.931	0.958
Prior-2 (Common Flat)	0.940	0.928	0.951

Conclusion:

The Bayesian framework achieves near-perfect classification accuracy (Mean AUC ≈ 0.94). Results are robust to the choice of prior (informative vs. flat). Demonstrates the framework’s capability for machine learning tasks on PDs for challenging real-world data.

References

Maroulas, Vasileios, Farzana Nasrin, and Christopher Oballe. 2020. “A Bayesian Framework for Persistent Homology.” SIAM J. Math. Data Sci. 2 (1): 48–74.

A Bayesian Framework for Persistent Homology

Contents

Original Paper

Introduction

Real World Problem

Real World Problem: Body Centered Cubic

Real World Problem: Face Centered Cubic

TDA Motivation

TDA Motivation

Background

Example Data

Persistence Diagrams

Simplices: Geometric Building Blocks

Simplicial Complexes

Example

Vietoris-Rips

VR Filtration

VR Visualization

Persistence Diagrams (PDs)

Persistence Diagrams

The Poisson Point Process (PPP)

Marked Poisson Point Processes

Bayesian Framework

Setting the Stage: Representation & Goal

Bayesian Framework: Analogy

The Model: Latent PD \(\mathcal{D}_X\)

The Model: Observed PD \(\mathcal{D}_Y\)

Bayes’ Theorem for PDs

Interpretation & Computation: Gaussian Mixtures

Worked Example

Parameters Summary

Simulated Point Clouds

Prior Intensities

Case I Results

Case II Results

Case III Results

Case IV Results

Classification

Real-World Example: Classifying Crystal Lattices

Bayesian Classification: Method & Results

Bayesian Classification: Method & Results

References

References